On-Shell Supermultiplets and Supersymmetric Ward Identities

On-shell superspace packages the physical helicities of a massless supermultiplet into a polynomial in Grassmann variables. Supercharges then act by multiplication and differentiation, so many component Ward identities become two linear equations on one superamplitude. Those equations relate components and expose forbidden sectors, but they do not determine the remaining kinematic functions or replace amplitude dynamics.

Required background. Massive and massless unitary supermultiplets supplies the physical helicity pairs. On-shell states and little-group scaling supplies spinor-helicity weights and the site’s all-outgoing external-state convention.

Helpful background. Spinor-helicity variables supplies brackets and complex three-point kinematics. Localized transformations and Ward–Takahashi identities supplies the current-to-amplitude logic.

One-particle on-shell superspace

Work first with complexified four-dimensional massless kinematics,

$$p_{\alpha\dot\alpha}=\lambda_\alpha\widetilde\lambda_{\dot\alpha}, \qquad \lambda\mapsto t\lambda, \qquad \widetilde\lambda\mapsto t^{-1}\widetilde\lambda.$$

The tilde on $\widetilde\lambda$ is an analytic dotted-spinor label. It becomes Hermitian conjugation only after a physical real Lorentzian boundary value and its energy-sign convention have been chosen.

Let $e_h(p)$ denote a dual external wavefunction of helicity $h$ in an all-outgoing amplitude, so $e_h\mapsto t^{-2h}e_h$. It is not a Hilbert-space ket, which would carry the inverse little-group weight. Introduce one Grassmann coordinate for $\mathcal N=1$ and set

$$\eta\mapsto t^{-1}\eta, \qquad \Omega_h(p,\eta) =e_h(p)+\eta\,e_{h-1/2}(p).$$

Both terms then have weight $t^{-2h}$. This is a generating function for two physical external states, not an off-shell superfield on spacetime.

CPT conjugation and Grassmann Fourier transformation are two separate operations. CPT first gives an independent conjugate wavefunction

$$\overline\Omega_{-h}(p,\bar\eta) =e_{-h}(p)+\bar\eta\,e_{-h+1/2}(p).$$

Choose $\bar\eta\mapsto t\bar\eta$ and $R(\bar\eta)=+1$, complementary to $\eta\mapsto t^{-1}\eta$ and $R(\eta)=-1$. We use the Berezin normalization and measure assignments

$$\int d\bar\eta\,\bar\eta=1, \qquad d\bar\eta\mapsto t^{-1}d\bar\eta, \qquad R(d\bar\eta)=-1.$$

One may then Fourier-transform the independent conjugate variable into the same $\eta$ representation:

$$\widetilde\Omega_{-h+1/2}(p,\eta) =\int d\bar\eta\,e^{\bar\eta\eta} \overline\Omega_{-h}(p,\bar\eta) =e_{-h+1/2}(p)+\eta\,e_{-h}(p).$$

The kernel $e^{\bar\eta\eta}$ is invariant, while the measure lowers the total little-group weight and R-charge by one. The transformed package therefore has weight $t^{2h-1}$, as required for its leading helicity $-h+\tfrac12$. Thus the Fourier transform reverses which conjugate state is the constant term; it does not itself perform CPT. This two-step construction is explicit in Elvang, Huang, and Peng 2011, §§ 2.2–2.3, Eqs. (2.17) and (2.28)–(2.29).

Representative same-$\eta$ packages are

Physical multiplet	Polynomial	Constant term	$\partial_\eta$ term
chiral sector	$\psi^++\eta\phi$	positive-helicity fermion	scalar
CPT-conjugate chiral sector	$\bar\phi+\eta\psi^-$	conjugate scalar	negative-helicity fermion
vector sector $\Phi$	$g^++\eta\lambda^+$	positive-helicity gluon	positive-helicity gluino
vector sector $\Psi$	$\lambda^-+\eta g^-$	negative-helicity gluino	negative-helicity gluon

The superscripts label helicity, not electric charge. If the algebra convention is $R(Q)=-1$, then

$$R(\eta)=-1, \qquad R(\partial_\eta)=+1.$$

For example, $R(g^+)=0$ and $R(\lambda^+)=+1$ make $\Phi$ homogeneous with total R-charge zero, while $R(\lambda^-)=-1$ and $R(g^-)=0$ make $\Psi$ homogeneous with total R-charge $-1$. In a theory with matter, replace these values by the chosen anomaly-free or spurionic charges and record the external representation leg by leg.

We use all momenta outgoing to match the site’s amplitude volume. An all-incoming convention is obtained by crossing every leg. Although both conventions write $\sum_i p_i=0$, crossing also exchanges particle and antiparticle labels and requires a consistent spinor phase choice; changing only the sign of $p_i$ is not a complete convention change.

Supercharges as multiplication and differentiation

Use left Grassmann derivatives, fixed by

$$\frac{\partial}{\partial\eta_i}(\eta_j\eta_k) =\delta_{ij}\eta_k-\delta_{ik}\eta_j.$$

On one leg define normalized kinematic operators

$$q_\alpha=\lambda_\alpha\eta, \qquad \bar q_{\dot\alpha} =\widetilde\lambda_{\dot\alpha}\frac{\partial}{\partial\eta}.$$

They obey

$$\{q_\alpha,\bar q_{\dot\alpha}\} =p_{\alpha\dot\alpha}.$$

The physical supercharges in the four-dimensional $\mathcal N=1$ convention are $Q=\sqrt2\,q$ and $\bar Q=\sqrt2\,\bar q$, restoring $\{Q,\bar Q\}=2p$. The $t$ weights cancel separately in $q$ and $\bar q$, proving legwise little-group invariance; their free undotted and dotted indices establish their Lorentz types.

For $n$ all-outgoing legs,

$$\mathsf q_\alpha=\sum_{i=1}^n\lambda_{i\alpha}\eta_i, \qquad \bar{\mathsf q}_{\dot\alpha} =\sum_{i=1}^n\widetilde\lambda_{i\dot\alpha} \frac{\partial}{\partial\eta_i},$$

and hence

$$\{\mathsf q_\alpha,\bar{\mathsf q}_{\dot\alpha}\} =P_{\alpha\dot\alpha}, \qquad P_{\alpha\dot\alpha}=\sum_i p_{i\alpha\dot\alpha}.$$

For $\mathcal N$-extended supersymmetry one introduces $\eta_i^A$ and repeats the construction for every R-symmetry index $A$.

Ward identities and the momentum delta function

Suppose the vacuum is supersymmetric, the asymptotic charges exist, and the quantum symmetry has no anomaly. Then $[Q,S]=0$. Separate the universal momentum-conservation distribution from the stripped superamplitude,

$$\mathbf A_n =(2\pi)^4\delta^{(4)}(P)\,\mathscr A_n,$$

and impose the Ward identities on the support $P=0$:

$$\mathsf q_\alpha\,\mathscr A_n=0, \qquad \bar{\mathsf q}_{\dot\alpha}\,\mathscr A_n=0.$$

The component form was developed for helicity amplitudes in Grisaru and Pendleton 1977, pp. 81–92. Nair’s maximally supersymmetric construction supplied the historical on-shell-superspace precursor Nair 1988, pp. 215–218.

At generic kinematics where the angle spinors span a two-dimensional space, define

$$\delta^{(2)}(\mathsf q) \equiv\frac12\epsilon^{\alpha\beta} \mathsf q_\alpha\mathsf q_\beta =\sum_{i<j}\langle ij\rangle\eta_i\eta_j.$$

Multiplication by either component of $\mathsf q$ annihilates this degree-two polynomial. The left-derivative convention also gives the exact identities

$$\frac{\partial}{\partial\eta_i}\delta^{(2)}(\mathsf q) =\sum_j\langle ij\rangle\eta_j =-\lambda_i^\alpha\mathsf q_\alpha,$$

and therefore

$$\bar{\mathsf q}_{\dot\alpha}\delta^{(2)}(\mathsf q) =-P^\alpha{}_{\dot\alpha}\mathsf q_\alpha=0 \qquad(P=0).$$

The simple degree-two solution is consequently

$$\mathscr A_n =\delta^{(2)}(\mathsf q)\,F(\lambda,\widetilde\lambda).$$

Supersymmetry leaves $F$ undetermined. It must still have the required little-group weights, internal charges, permutation properties, mass dimension, locality or factorization behavior, and loop analytic structure. At higher Grassmann degree, additional invariant polynomials multiply independent coefficient functions; Ward identities reduce the basis rather than compute those functions. The direct $\mathcal N<4$ construction is given in Elvang, Huang, and Peng 2011, §§ 3 and 6; the maximally supersymmetric basis construction is given in Elvang, Freedman, and Kiermaier 2010, §§ 2–4.

The rank-one three-point exception

Complex massless three-point kinematics has two branches. On the angle-bracket branch, $[ij]=0$ and the angle spinors can have rank two, so $\delta^{(2)}(\mathsf q)$ is nonzero. On the square-bracket branch, $\langle ij\rangle=0$ and all $\lambda_i$ are proportional. Then $\delta^{(2)}(\mathsf q)$ vanishes identically and cannot represent the nonzero conjugate sector.

The degree-one polynomial

$$m_{123} =[12]\eta_3+[23]\eta_1+[31]\eta_2$$

is the appropriate supersymmetric invariant on that rank-one branch: the Schouten identity gives $\bar{\mathsf q}m_{123}=0$, while momentum conservation and proportionality of the $\lambda_i$ make $\mathsf q\,m_{123}=0$. This exception is essential at the handoff to three-point amplitudes and is the three-point seed of the general invariant $m_{rsi}$ in Elvang, Huang, and Peng 2011, § 6, Eqs. (6.6)–(6.7).

Extracting component relations

Expand a stripped superamplitude in a fixed increasing Grassmann order,

$$\mathscr A_n(\eta) =\sum_{S\subseteq\{1,\ldots,n\}} \eta_S A_S, \qquad \eta_S=\prod_{i\in S}^{\nearrow}\eta_i.$$

A component amplitude is obtained with the left derivatives assigned by the external polynomial on each leg, followed by $\eta=0$. Operator products act on the expression to their right. Thus, for $i<j$, the coefficient of $\eta_i\eta_j$ is

$$A_{ij} =\left.\partial_j\partial_i\mathscr A_n\right|_{\eta=0}, \qquad \partial_i\partial_j(\eta_i\eta_j)=-1.$$

For the simple solution above,

$$A_{ij}=\langle ij\rangle F.$$

The globally meaningful relation is the cross-multiplied identity

$$\langle k\ell\rangle A_{ij} =\langle ij\rangle A_{k\ell}.$$

Only away from zeros of the relevant brackets may this be written as $A_{ij}/\langle ij\rangle=A_{k\ell}/\langle k\ell\rangle$. The same ordered-derivative extraction is reviewed in Elvang and Huang 2015, Chapter 4.

A five-point N=1 Yang–Mills fixture

For a concrete component round trip, assign legs $2$ and $3$ to

$$\Psi=\lambda^-+\eta g^-$$

and the remaining legs to $\Phi=g^++\eta\lambda^+$. Import from the amplitude volume the color-ordered tree seed

$$\mathcal F^{\mathcal N=1}_{5,23} =\frac{\langle23\rangle^3} {\langle12\rangle\langle23\rangle\langle34\rangle \langle45\rangle\langle51\rangle} \,\delta^{(2)}(\mathsf q).$$

The standard $\Phi$–$\Psi$ ordering convention gives

$$\begin{aligned} A(g_1^+,g_2^-,g_3^-,g_4^+,g_5^+) &=-\partial_2\partial_3\mathcal F_{5,23},\\ A(g_1^+,\lambda_2^-,g_3^-,\lambda_4^+,g_5^+) &=-\partial_3\partial_4\mathcal F_{5,23},\\ A(g_1^+,g_2^-,\lambda_3^-,\lambda_4^+,g_5^+) &=+\partial_2\partial_4\mathcal F_{5,23}. \end{aligned}$$

The minus signs record how a left derivative passes the odd $\Psi$ packages. Differentiating only the universal Grassmann polynomial yields two identifiable Ward relations:

$$\begin{aligned} \langle23\rangle A(g_1^+,\lambda_2^-,g_3^-,\lambda_4^+,g_5^+) &=\langle34\rangle A(g_1^+,g_2^-,g_3^-,g_4^+,g_5^+),\\ \langle23\rangle A(g_1^+,g_2^-,\lambda_3^-,\lambda_4^+,g_5^+) &=-\langle24\rangle A(g_1^+,g_2^-,g_3^-,g_4^+,g_5^+). \end{aligned}$$

This reproduces Elvang, Huang, and Peng 2011, § 3.1, Eqs. (3.4)–(3.9). The Parke–Taylor denominator and its dynamics belong to the amplitude construction; this page uses that known seed only to verify supersymmetric packaging and component signs.

Scope and failure conditions

The construction assumes:

• four-dimensional massless asymptotic states;
• a declared all-outgoing convention, with crossing phases handled consistently;
• a fixed normalization of $Q$, $\eta$, and left Grassmann differentiation;
• little-group-covariant dual external wavefunctions;
• CPT completion and internal representation recorded leg by leg;
• a supersymmetric vacuum and well-defined asymptotic charges; and
• no anomaly or regulator violation of the Ward identity.

Massive on-shell superspace needs extra little-group indices. Spontaneously broken supersymmetry gives Goldstino Ward identities rather than the unbroken relations used here. Infrared-divergent gauge-theory amplitudes may require regulated, inclusive, or dressed observables. At loop level the algebraic Ward identities remain constraints when the regulator and renormalization preserve supersymmetry, but they do not fix branch cuts, rational terms, or subtraction data.

Common pitfalls

Giving a ket the amplitude’s little-group weight. A one-particle ket and an all-outgoing dual external wavefunction transform inversely. State which object the generating polynomial contains before assigning $t^{-2h}$.

Treating a Grassmann Fourier transform as CPT. CPT first conjugates the physical states and their internal quantum numbers. A subsequent Fourier transform merely writes that conjugate multiplet in the same-$\eta$ representation.

Calling $\delta^{(2)}(\mathsf q)$ an amplitude. It solves the generic rank-two degree-two Ward constraints. The coefficient function contains the dynamics, and the rank-one three-point branch needs a different invariant.

Exercises

1. Left-derivative signs

Starting from $\delta^{(2)}(\mathsf q)=\sum_{i<j}\langle ij\rangle\eta_i\eta_j$, prove

$$\partial_i\delta^{(2)}(\mathsf q) =\sum_j\langle ij\rangle\eta_j$$

and then derive $\bar{\mathsf q}_{\dot\alpha}\delta^{(2)}(\mathsf q)=-P^\alpha{}_{\dot\alpha}\mathsf q_\alpha$.

Solution

For $j>i$, differentiating $\eta_i\eta_j$ gives $+\eta_j$. For $j<i$, the term is $\langle ji\rangle\eta_j\eta_i$ and the left derivative gives $-\langle ji\rangle\eta_j=+\langle ij\rangle\eta_j$. Multiplying by $\widetilde\lambda_{i\dot\alpha}$ and summing gives $-P^\alpha{}_{\dot\alpha}\mathsf q_\alpha$ after raising the undotted index with $\epsilon^{\alpha\beta}$. It vanishes on $P=0$.

2. Two component Ward relations

Apply the three displayed projection operators to $\mathcal F^{\mathcal N=1}_{5,23}$ and recover both cross-multiplied amplitude relations.

Solution

Write $\mathcal F_{5,23}=K\sum_{a<b}\langle ab\rangle\eta_a\eta_b$, where $K$ is independent of $\eta$. The ordered left derivatives and the displayed package signs give

$$A(g^+,g^-,g^-,g^+,g^+)=K\langle23\rangle,$$ $$A(g^+,\lambda^-,g^-,\lambda^+,g^+)=K\langle34\rangle, \qquad A(g^+,g^-,\lambda^-,\lambda^+,g^+)=-K\langle24\rangle.$$

Eliminating $K$ gives the two claimed identities without dividing by a bracket that might vanish.

3. The three-point rank drop

On the square-bracket branch, show that $\delta^{(2)}(\mathsf q)=0$ but $m_{123}$ is annihilated by both total supercharges.

Solution

Because all $\langle ij\rangle$ vanish, the explicit quadratic expansion of $\delta^{(2)}(\mathsf q)$ is zero. Schouten gives

$$\bar{\mathsf q}_{\dot\alpha}m_{123} =\widetilde\lambda_{1\dot\alpha}[23] +\widetilde\lambda_{2\dot\alpha}[31] +\widetilde\lambda_{3\dot\alpha}[12]=0.$$

Momentum conservation with proportional $\lambda_i$ makes $\mathsf q_\alpha$ proportional to the same Grassmann-linear combination $m_{123}$, so multiplication gives $\mathsf q_\alpha m_{123}\propto m_{123}^2=0$.

References

• Henriette Elvang, Daniel Z. Freedman, and Michael Kiermaier, “Solution to the Ward Identities for Superamplitudes,” Journal of High Energy Physics 2010 (2010), 103, DOI.
• Henriette Elvang and Yu-tin Huang, Scattering Amplitudes in Gauge Theory and Gravity, Cambridge University Press (2015), Chapter 4, DOI.
• Henriette Elvang, Yu-tin Huang, and Cheng Peng, “On-Shell Superamplitudes in $\mathcal N<4$ SYM,” Journal of High Energy Physics 2011 (2011), 031, DOI, arXiv.
• Marcus T. Grisaru and Hugh N. Pendleton, “Some Properties of Scattering Amplitudes in Supersymmetric Theories,” Nuclear Physics B 124 (1977), 81–92, DOI.
• V. P. Nair, “A Current Algebra for Some Gauge Theory Amplitudes,” Physics Letters B 214 (1988), 215–218, DOI.